In honor of the Abel Prize Committee’s decision to award credit for the celebrated conjecture on modularity of elliptic curves to Shimura, Taniyama, and Weil (in that order) in the course of awarding the Abel Prize to Andrew Wiles, I am publishing here for the first time an excerpt from the text of my talk, entitled *Mathematical Conjectures in the Light of Reincarnation*, at the conference *Two Streams in the Philosophy of Mathematics* that took place in 2009. (The remainder of the talk was reworked and expanded into Chapter 7 of *MWA.*) The conference was organized by David Corfield and Brendan Larvor and was held on the campus of the University of Hertfordshire in Hatfield, England. There are no photographic records of the conference, which left practically no internet trace whatsoever, apart from the program posted on the FOM website, so I have included a Wikimedia commons photo of the *Hatfield Galleria Shopping Mall*, which is where the conference dinner was held, across a traffic circle (*roundabout*) from the university campus.

For years I have wanted to write a comprehensive article about the controversy over the name of the conjecture for a philosophy of mathematics journal. But I have never had the patience to organize the themes of the controversy, and instead, as a way of relieving my persistent irritation with the way the controversy has been addressed, I have been inserting cranky fragments of arguments into articles and presentations where they don’t necessarily belong. Here is an example from an October 2009 draft that is actually called *PHILOSOPHICAL IRRITATION*. The original reason for my irritation is explained in the second paragraph, and it goes back to Serge Lang’s earliest interventions on behalf of Shimura, before Wiles proved Fermat’s Last Theorem; the words “chagrin” and “avalanche” allude to an incident that took place some five years after Wiles’s announcement, about which, perhaps, more will be written later.

Even before the Science Wars erupted, I had observed with increasing distress a bitter debate over the appropriate nomenclature for a conjecture of fundamental importance for my own work in number theory. As a graduate student I had been taught to refer to the conjecture as the “Weil conjecture” but a few years later, after a series of consultations I have not attempted to reconstruct among senior colleagues, the name was changed to “Taniyama-Weil conjecture.” By the beginning of my career, which fortuitously coincided with the official consecration, in the form of the four-week Corvallis summer school, of the Langlands program, this name designated this program’s iconic prediction and unattainable horizon, though even at the time it was understood to hold this position only as an effect of convergence after the fact, since Langlands had found his way to his program by another route, and the conjecture was primarily iconic for number theorists.[1] About ten years later the conjecture underwent another promotion when it was discovered[2] that it implied Fermat’s Last Theorem as a consequence. Serge Lang then began an energetic campaign (some details are recorded in the

AMS Notices) to change the name on the grounds that, after an initial hesitant formulation by Taniyama, it had been proposed in a more precise form by Shimura, the first to suggest the idea to Weil who, after a period of skepticism, not only published the first paper on the conjecture but wrote both Taniyama and Shimura out of the story. At this point the name of the conjecture underwent several bifurcations: Shimura-Taniyama-Weil for those inclined to generosity (and alphabetical order), Taniyama-Shimura-Weil for those with a certain view of history, Taniyama-Shimura or Shimura-Taniyama for those in Lang’s camp (including Shimura himself, as I later learned to my chagrin) who saw Weil as a treacherous interloper, and occasionally Taniyama-Weil for those who took pleasure in baiting Shimura. An often acrimonious exchange of opinions on the question, which enjoyed a brief revival after Wiles proved the conjecture in sufficient generality to imply Fermat’s Last Theorem and Taylor and his collaborators proved it in complete generality, has led to the current impasse where there is still no consensus on what the conjecture (now a theorem) should be called: French sources generally include Weil’s name, whereas many if not most American authors do not[3].What I found and continue to find most disheartening is that none of the quarrel’s protagonists saw fit to provide any guidance to resolving similar conflicts in the future. Indeed, with the exception of a relatively late[4] contribution by Serre, to which I return below, no one acknowledged that it might be of interest to consider the dispute as other than

sui generis, and the discussion remained largely in the forensic mode initiated by Lang. How and to whom to attribute ideas — a more or less isolated conjecture, a research program (such as the Langlands program), or akey stepin a proof — is the question of most moment in the development of individual careers, the distribution of power and resources, or the evolution of the self-consciousness of a branch of mathematics. It can be fruitfully analyzed by historical or sociological methods, specifically byscience studiesin one or another of its incarnations. It can also be given the status of a philosophical question. Indeed, the claim that such a question has philosophical content beyond what is accessible by history or sociology — that it can in some sense be analyzed in terms of principles whose nature remains to be determined — is itself a philosophical claim, and one that is likely to be contested. When such a question arises I would like to be able to answer it on the basis of principled arguments and not by joining a transitory alliance oractor network. It is not the sort of problem that typically appeals to philosophers.The problems about mathematics that do appeal to philosophers, according to the Oxford Handbook[5], include (for example)

- What, if anything, is mathematics about?
- …how do we know mathematics [if we do]?
- To what extent are the principles of mathematics objective and independent of…?
and so on. I see no room on this list for an account of how to attribute authorship to a conjecture. The word “conjecture” does not even appear in the index of this 800-page handbook (“theorem” occurs, but sparsely; more popular index entries are “truth,” “proof,” “proposition,” and “sentence,” as well as topics like “arithmetic,” “geometry,” and “number”). No guidance is forthcoming from the handbook as to whether a conjecture is ontological rather than epistemological or methodological. A conjecture must be a matter of importance to mathematicians, though, if so many of them, and not only the rival claimants and their friends, are willing to sacrifice valuable working time for fruitless belligerence in order to arrive at an accurate attribution.

It may just be that the conjecture is something the working mathematician is forced to cite by name, and that one attempts to trace it back to its original source in order to avoid irritating short-tempered colleagues. I briefly thought I had found a way to evade responsibility by referring to it as “the conjecture associated with the names of” followed by three names, a formulation that is undoubtedly objectively true (and admitting of ostensive proof in this very text); and at least one colleague followed me[6] along this deceptively innocuous path until the day we walked into an avalanche. Don’t follow my lead in suggesting that the problem can be evacuated into a matter of typographical convenience: do we know what kinds of “something” we can be “forced” to treat in this manner, and just what is this “force” that holds us in its grip? As of this writing many colleagues have given up on nominal attribution altogether and refer to the conjecture by what it says (the “Modularity Conjecture for elliptic curves,” for example). But how can a conjecture “say” anything?[7]

[1] Langlands’s own priorities were elsewhere, as he has frequently pointed out. His insights have been so influential in so many branches of mathematics that he can hardly be said to own the Langlands program any more. But he certainly has been clear and consistent about his own reasons for formulating the program that bears his name.

[2] By Frey, Serre, and Ribet. The word “discovered” is used here in the colloquial sense and expresses no philosophical commitment.

[3] I haven’t checked Japanese practice.

[4] Its publication came late in the story, but it’s clear from discussions with French colleagues that Serre had expressed his opinions on the matter quite early on.

[5] Stewart Shapiro, Chapter 1, p. 5; letters mine.

[6] Langlands used a similar formulation in a recent article in

Pour la Sciencebut I have no reason to think he didn’t come up with it on his own.[7] This at least looks more like a

Handbookquestion — but which one?

By the end of this text my irritation has expanded from the “short-tempered colleagues” who had promoted the controversy to include, uncharitably, the contributors to the *Handbook* who didn’t think to include material that would help to clarify what, if anything is really at stake (beyond personal antipathies, which should not be underestimated) in controversies of this kind.

Serre’s “relatively late” contribution to the debate is quite interesting, and has influenced a number of colleagues, but it looks like I never got around to revising the draft to keep the promise to explain what Serre added to the discussion. Maybe I will on this blog, or maybe I’ll actually write that philosophical article.

Akshay VenkateshIn a letter of Shimura to Taylor (published in his autobiography) he observes “There is another strange fact: nobody has asked me why I made the conjecture. Perhaps it looks so obvious now.”

I was reminded of this letter recently, when a physicist asked me this very question (i.e., how did people arrive at this conjecture)?

Shimura’s autobiography contains another letter to Taylor recounting his thought process. The two pieces of evidence were, apparently, that modularity is stable under quadratic twists, and that factors of the Jacobian of a Shimura curve are non-obviously modular (Eichler).

One has to admire the boldness of envisioning the modularity of all elliptic curves based on this!

Anyway, I agree with what I think you are saying: such conjectures form a core part of what mathematics is, but we don’t always acknowledge this.

LikeLike

mathematicswithoutapologiesPost authorMy main concerns in this passage are that, on the one hand, philosophy of mathematics — especially the mainstream (which was underrepresented at the conference in Hatfield) — makes no provisions for conjectures, or for mathematical ideas more generally; and that when mathematicians argue, often bitterly, about priority claims, they fail to recognize the need for criteria to evaluate such claims.

LikeLike

Akshay VenkateshI don’t know much about the philosophy of mathematics, but it seems to me that mathematics itself could be more open to valuing conjecture (not that we don’t value them, but I think there is, as Shimura is noting above, a tendency to take them as “naturally occurring resources”).

In this direction, I was struck that Subhash Khot won a Nevanlinna prize (and other awards) for the unique games conjecture some years ago. Those prizes are for the whole body of work around the UGC, of course, not just for a naked conjectural statement; but I think that this still reflects a more open attitude to conjecture than our current one in mathematics.

LikeLike

mathematicswithoutapologiesPost authorThe expression “mathematics itself could be more open” is philosophically very fruitful. If you mean by that “those who exercise power in mathematics could be more open” then I see the makings of a very lively exchange. Pierre Colmez just sent me a few excerpts from the Serre-Tate correspondence on these very questions — explicitly, on what the conjecture should be called, and why; implicitly, on who should get to decide. I’ll quote those excerpts, along with Serre’s “philosophical” criterion, in a later post.

LikeLike

Pingback: “Je m’en f…,” he wrote | Mathematics without Apologies, by Michael Harris